American Journal of Engineering Research (AJER) 2017 American Journal of Engineering Research (AJER) e-ISSN: 2320-0847 p-ISSN : 2320-0936 Volume-6, Issue-8, pp-01-07 www.ajer.org Research Paper Open Access Comparison on Fourier and Wavelet Transformation for an ECG Signal Mahamudul Hassan Milon (Department of Mathematics, Khulna University, Bangladesh) Corresponding Author: Mahamudul Hassan Milon ABSTRACT: Wavelet analysis is a new method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, signal processing, image processing, pattern recognition, computer graphics, the detection of aircraft and submarines and other medical image technology. Wavelets allow complex information such as music, speech, images and patterns to be decomposed into elementary forms at different positions and scales and subsequently reconstructed with high precision. Signal transmission is based on transmission of a series of numbers. The series representation of a function is important in all types of signal transmission. The wavelet representation of a function is a new technique. Wavelet transform of a function is the improved version of Fourier transform because Fourier transform is a powerful tool for analyzing the components of a stationary signal. But it is failed for analyzing the non stationary signal where as wavelet transform allows the components of a non-stationary signal to be analyzed. In this study our main goal is to compare an ECG signal for Fourier transformation and Wavelet transformation. Keywords: ECG Signal, Fourier transformation, Wavelet transformation, Haar Wavelet transform. ----------------------------------------------------------------------------------------------------------------------------- ---------- Date of Submission: 19-07-2017 Date of acceptance: 01-08-2017 ----------------------------------------------------------------------------------------------------------------------------- ---------- I. INTRODUCTION Historically, the concept of "Ondelettes" or "Wavelets" started to appear more frequently only in theearly 1980’s.One of the main reasons for the discovery of wavelets and wavelet transforms is that the Fourier transform does not contain the local information of signals. So the Fourier transform cannot be used for analyzing signals in a joint time and frequency domain. In 1982, Jean Morlet, in collaboration with a group of French engineers, first introduced the idea of wavelets as a family of functions constructed by using translation and dilation of a single function, called the mother wavelet, for the analysis of nonstationary sgnals. However this new concept can be viwed as the synthesis of various ideas originating from different disciplines including mathematics (Calderon-Zygmund operators and Littlewood-Paley theory), physics (coherent states in quantum mechanics and the renormalization group), and engineering (quadratic mirror filters, sideband coding in signal processing, and pyramidal algorithms in image processing)[1,2]. The Wavelets conference series was started by Andrew Laine in 1993. Under the leadership of Laine, Michael Unser, and Akram Aldroubi, it grew to be the leading venue for the dissemination of research on wavelets and their applications. Manos Papadakis replaced Akram Aldroubi as a conference chair in 2005, and since then the remainder of the leadership team has turned over, with the addition of Dimitri Van De Ville and Vivek Goyal[5,6]. Wavelets are mathematical functions that cut up data into different frequency components and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. The wavelet representation of a function is a new technique and it does not loss time information. Keeping these things in mind, our main goal in this thesis has been to provide both a systematic exposition of the basic ideas and results of wavelet transforms and some applications in time- frequency signal analysis. www.ajer.org Page 1 American Journal of Engineering Research (AJER) 2017 II. FOURIER TRANSFORM The Fourier transform is probably the most widely applied signal processing tool in science and engineering. It reveals the frequency composition of a time series x (t ) by transforming it from the time domain into the frequency domain. In 1807, the French mathematician Joseph Fourier found that any periodic signal can be presented by a weighted sum of a series of sine and cosine functions. However, because of the uncompromising objections from some of his contemporaries such as J. L. Lagrange (Herivel 1975), his paper on this finding never got published, until some 15 years later, when Fourier wrote his own book, The Analytical Theory of Heat (Fourier 1822). In that book, Fourier extended his finding to periodic signals, stating that an periodic signal can be represented by a weighted integral of a series of sine and cosine functions [5]. Such an integral is termed the Fourier transform. Using the notation of inner product, the Fourier transform of a signal can be expressed as i 2 ft i 2 ft X ( f ) x , e x (t )e dt (1) Assuming that the signal has finite energy, 2 x (t ) dt (2) Accordingly, the inverse Fourier transform of the signal x (t ) can be expressed as i 2 ft (3) x (t ) X ( f )e df Signals obtained experimentally through a data acquisition system are generally sampled at discrete time intervals T , instead of continuously, within a total measurement time T. Such a signal, defined as x k , can be transformed into the frequency domain by using the discrete Fourier transform (DFT), defined as N 1 1 k i 2 f k T DFT ( f ) x e n (4) n N k 0 Every signal can be written as a sum of sinusoids with different amplitudes and frequencies. Fig.1: Discrete Fourier transforms to transform data from time into the frequency domain III. WAVELET TRANSFORM A wave is usually defined as an oscillating function of time or space, such as a sinusoid. Fourier analysis is wave analysis. It expands signals or functions in terms of sinusoids (or, equivalently, complex exponentials) which has proven to be extremely valuable in mathematics, science, and engineering, especially for periodic, time-invariant, or stationary phenomena [7]. A wavelet is a "small wave", which has its energy concentrated in time to give a tool for the analysis of transient, non-stationary, or time-varying phenomena. It still has the oscillating wave-like characteristic but also has the ability to allow simultaneous time and frequency analysis with a flexible mathematical foundation. Fig.2: A Wave and A Wavelet www.ajer.org Page 2 American Journal of Engineering Research (AJER) 2017 The wavelet means small waves and in brief, a wavelet is an oscillation that decays quickly. Equivalent mathematical conditions for wavelet are: 2 i) ( x ) dx R ii ) ( x )dx 0 R 2 ˆ ( ) iii ) d R where ˆ ( ) is the Fourier Transform of ( x ) . IV. HAAR WAVELET TRANSFORM The Hungarian mathematician Alfred Haar first introduced the Haar function in 1909 in his Ph.D. thesis. A function defined on the real line as 1 1 for t 0 , 2 1 (5) (t ) 1 for t ,1 2 0 otherwise is known as the Haar function. The Haar function (t ) is the simplest example of a Haar Wavelet. The Haar function (t ) is a wavelet because it satisfies all the conditions of wavelet. This fundamental example has all the major features of the 1 general wavelet theory. Haar wavelet is discontinuous at t 0 , ,1 and it is very well localized in the time 2 domain [12]. The Fourier transform of (t ) is given by 2 sin ˆ ( ) i 4 i exp 2 4 2 sin 4 Re{ ˆ ( )} sin . 2 4 The graphs of Haar wavelet (t ) and its Fourier transform ( ) are shown in Fig.3 (a) and Fig.3 (b). Fig.3 (a): Haar wavelet Fig.3 (b): Fourier transform of Haar wavelet www.ajer.org Page 3 American Journal of Engineering Research (AJER) 2017 4.1 Haar Scaling Function The Haar scaling function can be defined as 1 if 0 t 1 (t ) 0 ,1 (t ) 0 otherwise . 4.2 Haar Wavelet Function Haar wavelet function can be written as (t ) (t ) (t ). 1 1 0 , ,1 2 2 4.3 Haar Wavelet Series and Wavelet Co-efficients If f is defined on [0, 1], then it has an expansion in terms of Haar functions as follows. Given any integer J 0 , j j 2 1 2 1 f (t ) f , (t ) f , (t ) J , K J , K j ,k j ,k k 0 j J k 0 J j 2 1 2 1 c (t ) d (t ) (6) J , K J , K j ,k j ,k k 0 j J k 0 The series (6) is known as the Haar wavelet series for f . d j , k and c j , k are known as the Haar wavelet co- efficients and the Haar scaling co-efficients respectively. 4.4 Advantages of Haar Wavelet Transform The Haar wavelet transform has a number of advantages: It is conceptually simple. It is fast. It is memory efficient, since it can be calculated in place without a temporary array. It is exactly reversible without the edge effects that are a problem with other wavelet transforms. 4.5 Comparison on Wavelet Transform and Fourier Transform The wavelet transform is often compared with the Fourier transform. Fourier transform is a powerful tool for analyzing the components of a stationary signal (a stationary signal is a signal where there is no change the properties of signal). For example, the Fourier transform is a powerful tool for processing signals that are composed of some combination of sine and cosine signals (sinusoids). The Fourier transform is less useful in analyzing non-stationary signal (a non-stationary signal is a signal where there is change the properties of signal). Wavelet transforms allow the components of a non- stationary signal to be analyzed.